190 



FRESNEL. 



rived at the result, setting out from the ideas he had 

 adopted of the nature of light. And, lastly, Newton de- 



l /', described with the velocities vvt, which are in a constant ratio to 





each other; and in times which will be Then, on the prin- 



v v> 



ciple of "least time," the condition is, 



-- -- ^minimum: 

 v vi 



or, differentiating and multiplying by v v>, 



V dl + v dl' = . . . . (1). 



Then if x be the surface of the medium, taking equal increments 

 d x on each side of the point of incidence, and dropping perpendiculars 



I, 



to give corresponding increments dldl',i and r being the angles of 

 incidence and refraction, we have geometrically 



sin i ,,. sin r 



, . . . (2); 



\M V j 



ax 

 and substituting in (1) it becomes 



ax 



or 



sin i v sin r = 0, 



v . 

 sm i =sin r. 



But, as i is necessarily greater than r, it follows that the v must be 

 greater than vf : or the law of the sines fulfils the condition of " least 

 time " on the wave theory. 



On the other hand, the principle of " least action" requires, instead 

 of equation (1), that we have 



I v-\-V v' = minimum, 

 or vdl+vldl'=Q: 



whence, by precisely the same process, there results 



v> . 

 sin t = sin r ; 



v 



